I had just solved this problem of Sierpinski
Prove that the sequence $2^n - 3, \; n = 2,3,4, \dots$ contains infinitely many terms divisible by $5$ and $13$ but no terms divisible by $5\cdot 13$.
For showing the existence of infinite terms divisible by $5$ and $13$ I basically applied Fermat's little theorem and some congruences.
Now for showing whether a term exists such that it is divisible by $65$ I thought of using the fact that if $a|b$ and $c | b$ and $gcd(a,c) = 1$ then $ac|b$. But for that, I would require to show that there is a common term divisible by both $5$ and $13$.
I finally found that values of $n$ satisfying $3 + 4k, \; k \in \mathbb{N}$ are divisible by $5$ and values of $n$ satisfying $4 + 12m, \; m \in \mathbb{N}$ are divisible by $13$. So now I have to prove that no $k$ and $m$ exist such that $3 + 4k = 4 + 12m$. This comes out to be:
$$4k - 12m = 1$$
Which is a linear diophantine equation. Now $gcd(4,12) \not\mid 1$ so no solution exists.
But I feel that doing it this way is wrong as $k, m \in \mathbb{N}$ but due to the $-$ (minus) sign in the equation negative values of $k$ and $m$ are also being considered.
So now coming to my main question, when are diophantine equations $ax + by = c$ and $ax - by = c$ equivalent and what are their differences?
P.S I did solve this without using the diophantine part by proving that no positive integers $k$ and $m$ exist such that $k - 3m = \frac{1}{4}$.
No need to worry about signs $4k-12m$ is a multiple of $\gcd(4,12)=4$, while $1$ is not.
Edit: Let$a,b>0$.
Then, $ax+by=c$ is practically the same as $ax-by=c$ if $x,y\in\mathbb{Z}$ because if $(x',y')$ is a solution of the first one, then $(x',-y')$ is a solution for the other one.
Now, if you are just looking for $x,y\in\mathbb{N}$, then $ax+by=c$ will have a finite number of solutions, while $ax-by=c$ may have infinite solutions (note that if $(x',y')$ is a solution then $(x'+kb,y'+ka)$ is a solution)